Abstract
The idea that the mass m of an elementary particle is explained in the semi-classical approximation by quantum-mechanical zero-point vacuum fluctuations has been applied previously to spin-1/2 fermions to yield a real and positive constant value for m, expressed through the spinorial connection Γ i in the curved-space Dirac equation for the wave function ψ due to Fock. This conjecture is extended here to bosonic particles of spin 0 and spin 1, starting from the basic assumption that all fundamental fields must be conformally invariant. As a result, in curved space-time there is an effective scalar mass-squared term \(m_{0}^{2}=-R/6=2\varLambda_{\mathrm{b}}/3\), where R is the Ricci scalar and Λ b is the cosmological constant, corresponding to the bosonic zero-point energy-density, which is positive, implying a real and positive constant value for m 0, through the positive-energy theorem. The Maxwell Lagrangian density \(\mathcal{L} =- \sqrt{-g}F_{ij}F^{ij}/4\) for the Abelian vector field F ij ≡A j,i −A i,j is conformally invariant without modification, however, and the equation of motion for the four-vector potential A i contains no mass-like term in curved space. Therefore, according to our hypothesis, the free photon field A i must be massless, in agreement with both terrestrial experiment and the notion of gauge invariance.
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Appendix: The Fermi Lagrangian of Electromagnetism
Appendix: The Fermi Lagrangian of Electromagnetism
The Lagrangian density of the electromagnetic field
can be rewritten in Minkowski space-time, where \(\sqrt{-g}=1\) and covariant derivatives reduce to partial derivatives, as
where the Fermi Lagrangian is defined by
Thus, expression (157) differs from \(\mathcal{L}_{\mathrm{F}}\) only by a total divergence, if the Lorenz gauge condition (104) is imposed, which is guaranteed by the condition (110). Expression (158) implies a non-vanishing canonical momentum
allowing construction of the Hamiltonian.
There is no obvious way of generalizing this procedure to curved space-time, however. For the covariant form of Eq. (158) is
and therefore we have to rewrite Eq. (156) as
The first term on the right-hand side of Eq. (161) is expression (160), but the second term cannot be expressed as the difference of the total divergence \(\frac{1}{2}(\sqrt{-g}A_{j}A^{i;j})_{,i}\) and the product of A j times the jth derivative of \(A^{i}_{;i}\), in particular because \(A^{i;j}_{;i}\neq A_{;i}^{i;j}\). Nor is any meaningful simplification achieved by imposing a metric gauge condition, for example, the de Donder–Lanczos condition
in addition to the electromagnetic gauge condition (104), due to the fact that expression (160) contains a term which is quadratic in both \(\varGamma^{i}_{jk}\) and A i .
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Pollock, M.D. On Vacuum Fluctuations and Particle Masses. Found Phys 42, 1300–1328 (2012). https://doi.org/10.1007/s10701-012-9668-2
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DOI: https://doi.org/10.1007/s10701-012-9668-2